Chapter 3.2, Problem 24E

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

(a) Suppose that *f* is differentiable on
*f*′ has at least one root.

(b) Suppose / is twice differentiable on
*f*′′ has at least one real root.

(c) Can you generalize parts (a) and (b)?

To determine

**(a)**

**To show:**

Explanation

**1) Concept:**

Using the Rolle’s Theorem verify the result

**2) Theorem:**

Rolle’s theorem- Let

1)

2)

3)

Then, there is number

[ if a and

**3) Given:**

**3) Calculation:**

By Rolle’s Theorem, there is a number

And

Therefore, the function

**Conclusion:**

Therefore,

**(b)**

**To show:**

**Solution:**

**1) Concept:**

Using the Rolle’s Theorem verify the result

**2) Theorem:**

a) Roll’s theorem- Let

1)

2)

3)

Then, there is number

[

means between two real roots of

**3) Given:**

**4) Calculation:**

Suppose

Therefore,

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